SET007 Axioms: SET007+919.ax
%------------------------------------------------------------------------------
% File : SET007+919 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Homeomorphisms of Jordan Curves
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : jordan24 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 24 ( 0 unt; 0 def)
% Number of atoms : 227 ( 10 equ)
% Maximal formula atoms : 20 ( 9 avg)
% Number of connectives : 222 ( 19 ~; 2 |; 104 &)
% ( 6 <=>; 91 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 10 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 39 ( 38 usr; 0 prp; 1-4 aty)
% Number of functors : 32 ( 32 usr; 8 con; 0-4 aty)
% Number of variables : 77 ( 72 !; 5 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_jordan24,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ? [B] :
( m1_relset_1(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
& v1_jordan24(B,A) ) ) ).
fof(cc1_jordan24,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_relset_1(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
=> ( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
& v1_jordan24(B,A) )
=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
& v5_pre_topc(B,k5_pcomps_1(A),k5_pcomps_1(A)) ) ) ) ) ).
fof(cc2_jordan24,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_relset_1(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
=> ( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
& v1_jordan24(B,A) )
=> ( v1_funct_1(B)
& v2_funct_1(B)
& ~ v1_xboole_0(B)
& v1_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
& v2_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
& v3_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
& v5_pre_topc(B,k5_pcomps_1(A),k5_pcomps_1(A))
& v1_partfun1(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
& v1_t_0topsp(B,k5_pcomps_1(A),k5_pcomps_1(A))
& v3_tops_2(B,k5_pcomps_1(A),k5_pcomps_1(A)) ) ) ) ) ).
fof(d1_jordan24,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ( r1_jordan24(A,B,C,D)
<=> ( r2_hidden(C,B)
& r2_hidden(D,B)
& ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
=> ( ( r2_hidden(E,B)
& r2_hidden(F,B) )
=> r1_xreal_0(k1_gobrd14(A,E,F),k1_gobrd14(A,C,D)) ) ) ) ) ) ) ) ) ) ).
fof(t1_jordan24,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ? [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
& ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& r1_jordan24(np__2,A,B,C) ) ) ) ).
fof(d2_jordan24,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
& m2_relset_1(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A))) )
=> ( v1_jordan24(B,A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& v3_vectmetr(C,A)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A))
& C = B ) ) ) ) ).
fof(d3_jordan24,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)))
& m2_relset_1(B,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2))) )
=> ( B = k1_jordan24(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),B,C) = k23_euclid(k3_complex1(k3_complex2(k2_xcmplx_0(k21_euclid(C),k3_xcmplx_0(k22_euclid(C),k1_xcmplx_0)),A)),k4_complex1(k3_complex2(k2_xcmplx_0(k21_euclid(C),k3_xcmplx_0(k22_euclid(C),k1_xcmplx_0)),A))) ) ) ) ) ).
fof(t2_jordan24,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__0,A)
=> ( r1_xreal_0(k4_real_1(np__2,k32_sin_cos),A)
| ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_pcomps_1(k14_euclid(np__2))),u1_struct_0(k5_pcomps_1(k14_euclid(np__2))))
& m2_relset_1(B,u1_struct_0(k5_pcomps_1(k14_euclid(np__2))),u1_struct_0(k5_pcomps_1(k14_euclid(np__2)))) )
=> ( B = k1_jordan24(A)
=> v1_jordan24(B,k14_euclid(np__2)) ) ) ) ) ) ).
fof(t3_jordan24,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> ( r1_jordan24(np__2,C,A,B)
=> r1_jordan24(np__2,k4_pre_topc(k15_euclid(np__2),k15_euclid(np__2),k2_jgraph_2(D,E,D,F),C),k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k2_jgraph_2(D,E,D,F),A),k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k2_jgraph_2(D,E,D,F),B)) ) ) ) ) ) ) ) ).
fof(t4_jordan24,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r1_xreal_0(np__0,D)
& r1_jordan24(np__2,C,A,B) )
=> ( r1_xreal_0(k4_real_1(np__2,k32_sin_cos),D)
| r1_jordan24(np__2,k4_pre_topc(k15_euclid(np__2),k15_euclid(np__2),k1_jordan24(D),C),k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k1_jordan24(D),A),k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k1_jordan24(D),B)) ) ) ) ) ) ) ).
fof(t5_jordan24,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k3_complex2(A,k1_real_1(B)) = k3_complex2(A,k5_real_1(k4_real_1(np__2,k32_sin_cos),B)) ) ) ).
fof(t6_jordan24,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k1_jordan24(k1_real_1(A)) = k1_jordan24(k5_real_1(k4_real_1(np__2,k32_sin_cos),A)) ) ).
fof(t7_jordan24,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ? [B] :
( m1_topgrp_1(B,k15_euclid(np__2))
& r1_jordan24(np__2,k4_pre_topc(k15_euclid(np__2),k15_euclid(np__2),B,A),k23_euclid(k1_real_1(np__1),np__0),k23_euclid(np__1,np__0)) ) ) ).
fof(d4_jordan24,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( l1_pre_topc(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v2_jordan24(C,A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_pre_topc(D,A)
=> v4_pre_topc(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ) ).
fof(t8_jordan24,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( v2_funct_1(C)
& v2_funct_2(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(C,A,B)
<=> v2_jordan24(C,A,B) ) ) ) ) ) ).
fof(t9_jordan24,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( k3_subset_1(A,B) = k1_xboole_0
<=> B = A ) ) ).
fof(t10_jordan24,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_connsp_1(D,A)
=> v2_connsp_1(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ) ).
fof(t11_jordan24,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r3_connsp_1(A,D)
=> r3_connsp_1(B,k4_pre_topc(A,B,C,D)) ) ) ) ) ) ) ).
fof(t12_jordan24,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_funct_1(k7_relat_1(C,D))
& v1_funct_2(k7_relat_1(C,D),u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(k3_pre_topc(B,k4_pre_topc(A,B,C,D))))
& m2_relset_1(k7_relat_1(C,D),u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(k3_pre_topc(B,k4_pre_topc(A,B,C,D)))) ) ) ) ) ) ).
fof(t13_jordan24,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v5_pre_topc(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(k3_pre_topc(B,k4_pre_topc(A,B,C,D))))
& m2_relset_1(E,u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(k3_pre_topc(B,k4_pre_topc(A,B,C,D)))) )
=> ( E = k7_relat_1(C,D)
=> v5_pre_topc(E,k3_pre_topc(A,D),k3_pre_topc(B,k4_pre_topc(A,B,C,D))) ) ) ) ) ) ) ) ).
fof(t14_jordan24,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(k3_pre_topc(B,k4_pre_topc(A,B,C,D))))
& m2_relset_1(E,u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(k3_pre_topc(B,k4_pre_topc(A,B,C,D)))) )
=> ( E = k7_relat_1(C,D)
=> v3_tops_2(E,k3_pre_topc(A,D),k3_pre_topc(B,k4_pre_topc(A,B,C,D))) ) ) ) ) ) ) ) ).
fof(t15_jordan24,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( r4_connsp_1(A,E,D)
=> r4_connsp_1(B,k4_pre_topc(A,B,C,E),k4_pre_topc(A,B,C,D)) ) ) ) ) ) ) ) ).
fof(t16_jordan24,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [B] :
( m1_topgrp_1(B,k15_euclid(np__2))
=> ( v2_jordan1(A)
=> v2_jordan1(k4_pre_topc(k15_euclid(np__2),k15_euclid(np__2),B,A)) ) ) ) ).
fof(dt_k1_jordan24,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( v1_funct_1(k1_jordan24(A))
& v1_funct_2(k1_jordan24(A),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)))
& m2_relset_1(k1_jordan24(A),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2))) ) ) ).
%------------------------------------------------------------------------------